MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imass1 Structured version   Visualization version   GIF version

Theorem imass1 6119
Description: Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
Assertion
Ref Expression
imass1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Proof of Theorem imass1
StepHypRef Expression
1 ssres 6021 . . 3 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
2 rnss 5950 . . 3 ((𝐴𝐶) ⊆ (𝐵𝐶) → ran (𝐴𝐶) ⊆ ran (𝐵𝐶))
31, 2syl 17 . 2 (𝐴𝐵 → ran (𝐴𝐶) ⊆ ran (𝐵𝐶))
4 df-ima 5698 . 2 (𝐴𝐶) = ran (𝐴𝐶)
5 df-ima 5698 . 2 (𝐵𝐶) = ran (𝐵𝐶)
63, 4, 53sstr4g 4037 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3951  ran crn 5686  cres 5687  cima 5688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698
This theorem is referenced by:  predrelss  6358  vdwnnlem1  17033  dprdres  20048  imasnopn  23698  imasncld  23699  imasncls  23700  utoptop  24243  restutop  24246  ustuqtop3  24252  utopreg  24261  metustbl  24579  imadifxp  32614  gsumfs2d  33058  esum2d  34094  eulerpartlemmf  34377  bj-imdirco  37191  brtrclfv2  43740  frege97d  43765  frege109d  43770  frege131d  43777  hess  43793  resimass  45246  setrecsss  49220
  Copyright terms: Public domain W3C validator